All verses are quoted from the NASB anything in [square brackets] is added by me for clarity.

In the gospel of Luke Jesus says,

"The Law and the Prophets were proclaimed until John; since that time the gospel of the kingdom of God has been preached, " Luke 16:16a

Here it sounds like Jesus is saying that the "Law and the Prophets" were once taught but now they are not being taught. But something entirely separate is being taught which is called the "Gospel of the Kingdom of God".

Verses

However we understand this verse, we can clearly see that Jesus is making a historical claim about teaching. Up until John the Baptist the "Law and the Prophets" [ie. the Old Testament] were taught. This is why even though John the Baptist is mentioned in the New Testament, he is considered the last of the Old Testament prophets. Let us see the history of what happened -

"Now when Jesus heard that John had been taken into custody, He withdrew into Galilee; and leaving Nazareth, He came and settled in Capernaum, which is by the sea, in the region of Zebulun and Naphtali. This was to fulfill what was spoken through Isaiah the prophet: `THE LAND OF ZEBULUN AND THE LAND OF NAPHTALI, BY THE WAY OF THE SEA, BEYOND THE JORDAN, GALILEE OF THE GENTILES - THE PEOPLE WHO WERE SITTING IN DARKNESS SAW A GREAT LIGHT, AND THOSE WHO WERE SITTING IN THE LAND AND SHADOW OF DEATH, UPON THEM A LIGHT DAWNED.' From that time Jesus began to preach and say, `Repent, for the kingdom of heaven is at hand.' " Matthew 4:12-17

Notice that it is after John the Baptist is taken into custody (v 12) that Jesus begins to preach the kingdom of heaven is at hand (v 17). Mark's gospel records the same detail for us, as we can read -

"Now after John had been taken into custody, Jesus came into Galilee, preaching the gospel of God, and saying, `The time is fulfilled, and the kingdom of God is at hand; repent and believe in the gospel.' " Mark 1:14-15

So the "Law and the Prophets" were proclaimed until John the Baptist was taken into custody by King Herod. And from that time onwards the Gospel of the Kingdom of God/Heaven is preached. Now the first full length sermon that Jesus preaches that is recorded for us, after John is taken into custody can be read in Matthew chapter 5 (the next chapter after John is taken into custody).

"Do not think that I [Jesus] came to abolish the Law or the Prophets; I did not come to abolish but to fulfill. For truly I say to you, until heaven and earth pass away, not the smallest letter or stroke shall pass from the Law until all is accomplished. Whoever then annuls one of the least of these commandments, and teaches others to do the same, shall be called least in the kingdom of heaven; but whoever keeps and teaches them, he shall be called great in the kingdom of heaven." Matthew 5:17-19

So during Jesus preaching he mentions that He did not come to abolish the Law of God. Moreover as Jesus continues preaching His "gospel of the Kingdom of God" message, He quotes from the Old Testament (the "Law and the Prophets"), in these verses -

"You have heard that the ancients were told, 'YOU SHALL NOT COMMIT MURDER' and 'Whoever commits murder shall be liable to the court.' " Matthew 5:21
"You have heard that it was said, 'YOU SHALL NOT COMMIT ADULTERY'; " Matthew 5:27
"It was said, 'WHOEVER SENDS HIS WIFE AWAY, LET HIM GIVE HER A CERTIFICATE OF DIVORCE'; " Matthew 5:31
"Again, you have heard that the ancients were told, 'YOU SHALL NOT MAKE FALSE VOWS, BUT SHALL FULFILL YOUR VOWS TO THE LORD.' " Matthew 5:33
"You have heard that it was said, 'AN EYE FOR AN EYE, AND A TOOTH FOR A TOOTH.' " Matthew 5:38
"You have heard that it was said, 'YOU SHALL LOVE YOUR NEIGHBOR' " Matthew 5:43a

and then He explains that not only is it valid in the physical but that there is also a spiritual dimension to it too. As He explains in the corresponding verses listed below -

"But I say to you that everyone who is angry with his brother shall be guilty before the court; and whoever says to his brother, 'You good-for-nothing,' shall be guilty before the supreme court; and whoever says, 'You fool,' shall be guilty enough to go into the fiery hell. " Matthew 5:22
"but I say to you that everyone who looks at a woman with lust for her has already committed adultery with her in his heart. " Matthew 5:28
"but I say to you that everyone who divorces his wife, except for the reason of unchastity, makes her commit adultery; and whoever marries a divorced woman commits adultery. " Matthew 5:32
"But I say to you, make no oath at all" Matthew 5:34a
"But I say to you, do not resist an evil person; but whoever slaps you on your right cheek, turn the other to him also. " Matthew 5:39
"But I say to you, love your enemies and pray for those who persecute you," Matthew 5:44

So Jesus' pattern is to say "You have heard it said" and then quote from the Old Testament and then He says "but I say to you" and He expands the meaning of the Old Testament passage. So in Jesus' gospel of the kingdom of God message, He preaches the Law and that it is even harder to keep than was previously thought. So since He is not abolishing the Law or the Prophets we can change our initial view of things to this -

If we re-read the original passage in Luke that started our investigation we see, in context it says -

"The Law and the Prophets were proclaimed until John; since that time the gospel of the kingdom of God has been preached, and everyone is forcing his way into it. But it is easier for heaven and earth to pass away than for one stroke of a letter of the Law to fail." Luke 16:16-17

So Jesus confirms that God's message is not being changed (God does not change His mind). The Law the whole Law still applies (that is after all why we need Jesus, if the Law was abolished then we would not need a Saviour). Again to point out that the gospel is compatible with the Law, Jesus then tells this parable in Luke 16 -

"Now there was a rich man, and he habitually dressed in purple and fine linen, joyously living in splendor every day. And a poor man named Lazarus was laid at his gate, covered with sores, and longing to be fed with the crumbs which were falling from the rich man's table; besides, even the dogs were coming and licking his sores. Now the poor man died and was carried away by the angels to Abraham's bosom; and the rich man also died and was buried. In Hades he lifted up his eyes, being in torment, and saw Abraham far away and Lazarus in his bosom. And he cried out and said, 'Father Abraham, have mercy on me, and send Lazarus so that he may dip the tip of his finger in water and cool off my tongue, for I am in agony in this flame.' But Abraham said, 'Child, remember that during your life you received your good things, and likewise Lazarus bad things; but now he is being comforted here, and you are in agony. And besides all this, between us and you there is a great chasm fixed, so that those who wish to come over from here to you will not be able, and that none may cross over from there to us.' And he said, 'Then I beg you, father, that you send him to my father's house - for I have five brothers--in order that he may warn them, so that they will not also come to this place of torment.' But Abraham said, 'They have Moses and the Prophets; let them hear them.' But he said, 'No, father Abraham, but if someone goes to them from the dead, they will repent!' But he said to him, 'If they do not listen to Moses and the Prophets, they will not be persuaded even if someone rises from the dead.' " Luke 16:19-20

Notice that the point of this parable is that people will not repent if they do not believe the Old Testament (Moses, who gave the Law; and the Prophets). They will not believe the gospel of the resurrected Jesus from the dead if they believe God was lying in the Old Testament. Thus the Law and Prophets (Old Testament) is compatible with the Gospel of the Kingdom of God (New Testament).